Transcript Management of Financial Risk

Computational Finance
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
CF-4
Bank Hapoalim
Jun-2001
Plan
1. Monte Carlo Method
2. Variance Reduction Methods
3. Quasi Monte Carlo
4. Permuting QMC sequences
5. Dimension reduction
6. Financial Applications
simple and exotic options
American type
prepayments
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slide 2
Monte Carlo
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slide 3
Monte Carlo
1
0.5
-1
-0.5
0.5
1
-0.5
-1
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Monte Carlo
1
0.5
-1
-0.5
0.5
1
-0.5
-1
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slide 5
Monte Carlo
1
0.5
-1
-0.5
0.5
1
-0.5
-1
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Speed of convergence
0.08
0.06
Whole circle
0.04
0.02
20
40
60
80
100
-0.02
-0.04
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Upper triangle
CF4
slide 7
Smart Sampling
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Spectral Truncation
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Variance Reduction
Let X() be an option.
Let Y be a similar option which is correlated
with X but for which we have an analytic
formula.
Introduce a new random variable
X  ( )  X ( )   Y ( )  Y 
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Variance Reduction
The variance of the new variable is
var[X  ]  var[X ]  2 cov[X , Y ]   var[Y ]
2
If 2cov[X,Y] > 2var[Y] we have reduced
the variance.
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slide 11
Variance Reduction
The optimal value of  is
cov[X , Y ]
 
var[Y ]
*
Then the variance of the estimator becomes:
var[ X  * ]  (1   XY )var[ X ]
2
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slide 12
Variance Reduction
Note that we do not have to use the optimal
* in order to get a significant variance
reduction.
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slide 13
Multidimensional Variance Reduction
A simple generalization of the method can be
used when there are several correlated
variables with known expected values.
Let Y1, …, Yn be variables with known means.
Denote by Y the covariance matrix of
variables Y and by XY the n-dimensional
vector of covariances between X and Yi.
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Multidimensional Variance Reduction
Then the optimal projection on the Y plane is
given by vector:  *   TXY Y1
The resulting minimum variance is
var[ X  * ]  (1  RXY )var[ X ]
2
where
RXY 
2
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 TXY Y1 XY
var[X ]
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slide 15
Variance Reduction
• Antithetic sampling
• Moment matching/calibration
• Control variate
• Importance sampling
• Stratification
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slide 16
Monte Carlo in Risk Management
• Distribution of market factors
• Simulation of a large number of events
• P&L for each scenario
• Order the results
• VaR = lowest quantile
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slide 17
How to design MC
The central point is to model the
distribution of relevant risk factors.
For example, in pricing you should use the
risk-neutral distribution.
For risk measurement use true distribution.
What should be used for an estimate of
frequency of hedge?
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slide 18
Geometrical Brownian Motion
dS  rSdt  SdB
option e
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 r (T t )
Erisk neutral  payoff(S )
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slide 19
Lognormal process
dS  rSdt  SdB
 
d (log S )   r 
2

2
S (t )  S (0)e
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 2
 r

2

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
dt  dB


 t Z N ( 0 ,1) t


slide 20
Euler Scheme
dS  rSdt  SdB
S  rSt  SZN (0,1) t  O(t )
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Milstein Scheme
dS  rSdt  SdB
S  rSt  SZN ( 0,1) t 
1 2 2 2
2
 S ( Z N ( 0,1)  1)t  O(t )
2
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MC for simple options
Needs["Statistics`NormalDistribution`"]
Clear[MCEuropean, MCEuropeanCall, MCEuropeanPut]
nor[mu_,sig_]:=Random[NormalDistribution[mu,sig]];
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MC for simple options
MCEuropean[s_, T_, r_, _, n_, exercise_Function]:=
Module[{m = N[Log[s]+(r - 0.52)*T], sg=N[ Sqrt[T] ], tbl},
tbl= Table[nor[m, sg], {i, n}];
Exp[-r*T]*Map[exercise, Exp[Join[tbl, 2*m - tbl]]]//
{Mean[#], StandardErrorOfSampleMean[#]}&
]
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slide 24
MC for simple options
MCEuropeanCall[s_, x_, T_, r_,  _, n_]:=
MCEuropean[s, T, r, , n, Max[#-x,0]&]
MCEuropeanPut[s_, x_, T_, r_,  _, n_]:=
MCEuropean[s, T, r, , n, Max[x-#,0]&]
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slide 25
MC for path dependent options
RandomWalk[n_Integer] :=
FoldList[Plus, 0, Table[Random[] - 1/2, {n}]];
ListPlot[ RandomWalk[500], PlotJoined -> True];
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slide 26
MC for path dependent options
The function paths generates a random sample of price
paths for the averaging period. It returns a list of
numberPaths random paths, each consisting of
numberPrices prices over the period from time T1 to time
T. The prices at the start of the period are given by the
appropriate lognormal distribution for time T1.
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slide 27
MC for path dependent options
paths[s_,sigma_,T1_,T_,r_,numberPrices_,numberPaths_]:=
Module[{meanAtT1=Log[s]+(r-sigma^2/2)*T1,
sigmaAtT1 = sigma*Sqrt[T1],
meanPath = 1+ r*(T-T1)/(numberPrices-1),
sigmaPath = sigma*Sqrt[(T-T1)/(numberPrices-1)]
},
Table[NestList[# nor[meanPath,sigmaPath]&,
Exp[nor[meanAtT1,sigmaAtT1]],
numberPrices - 1], {i,numberPaths}]
]
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MC for Asian options
MCAsianCall[s_,x_,sigma_,T1_,T_,r_,numberPrices_,numberPaths_]:=
Module[{ t1, t2, t3},
t1 = paths[s,sigma,T1,T,r,numberPrices,numberPaths] ;
t2 = Map[Max[0,Mean[#] - x]&, t1];
t3 = Exp[-T*r]*t2;
{Mean[t3], StandardErrorOfSampleMean[t3]}
]
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slide 29
Quasi Monte Carlo
• Van der Corput
• Halton
• Haber
• Sobol
• Faure
• Niederreiter
• Permutations
• Nets
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slide 30
Quasi Monte Carlo
Are efficient in low (1-2) dimensions.
Sobol sequences can be used for small
dimensions as well.
As an alternative one can create a fixed set of
well-distributed paths.
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slide 31
Do not use free sequences
1
0.8
0.6
0.4
0.2
0.2
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0.6
0.8
1
slide 32
Other MC applications
• Pricing
• Optimal hedging
• Impact of dividends
• Bounds on a basket
• Prepayments
• Tranches of MBS
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slide 33
Other MC related topics
Use of analytical approximations
Richardson extrapolation
Ratchets example
American properties
Bundling
Modeling Fat tails
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slide 34
Value of Value-at-Risk
Zvi Wiener
The Hebrew University of Jerusalem
[email protected]
CF-4
Bank Hapoalim
Jun-2001
VaR
1 day
1% probability 1w
1% probability 1d
1 week
P&L
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slide 36
100%
VaRA
1%
0.5%
-3
-2
-1
0
VaR is not sub-additive.
Cumulative distribution of assets A and B.
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slide 37
-3 0.005 -2 0.005 -1
0.00495
0.00495
0.99
0.9801
B
A
Total loss $1
0.99
-1
Total loss $2
0.000025
0.000025
0.00495
0.005
-2
0.000025
0.000025
0.00495
VaRA+B
0.005
-3
Joint distribution of A and B
The integral over the gray triangle is
0.99*0.99 + 0.99*0.005*2/2 = 0.98505, which means that
there are more than 1% chances to be outside the
gray area and VaRA+B > VaRA+VaRB=$2.
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slide 38
Model
• Bank’s choice of an optimal system
• Depends on the available capital
• Current and potential capital needs
• Queuing model as a base
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Required Capital
Let A be total assets
C – capital of a bank
 - percentage of qualified assets
k – capital required for traded assets
C  A(1   )0.08  Ak
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Maximal Risk (Assets)
Amax
C

(1   )0.08  k
The coefficient k varies among systems, but a better
(more expensive) system provides more precise risk
measurement, thus lower k.
Cost of a system is p, paid as a rent (pdt during dt).
Amax is a function of C and p.
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slide 41
Risky Projects
Deposits arrive and are withdrawn randomly.
All deposits are of the same size.
Invested according to bank’s policy.
Can not be used if capital requirements are not
satisfied.
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slide 42
Arrival of Risky Projects
We assume that risky projects arrive randomly
(as a Poisson process with density ).
This means that there is a probability dt that
during dt one new project arrives.
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Arrival of Risky Projects
A new project is undertaken if the bank has
enough capital (according to the existing risk
measuring system).
We assume that one can NOT raise capital or
change systems quickly.
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CF4
slide 44
Termination of Risky Projects
We assume that each risky project disappears
randomly (as a Poisson process with density ).
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Termination of Risky Projects
We assume that each risky project disappears
randomly (as a Poisson process with density ).
This means that there is a probability ndt that
during dt one out of n existing projects
terminates.
With probability (1-ndt) all existing projects
will be active after dt.
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CF4
slide 46
Profit
We assume that each existing risky project gives
a profit of dt during dt.
Thus when there are n active projects the bank
has instantaneous profit (n-p)dt.
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CF4
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States
After C and p are chosen, the maximal number
of active projects is given by s=Amax(C,p).




2
s
0
1
2
s-1
s
0
1
2
s-1
s
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States




2
s
0
1
2
s-1
s
0
1
2
s-1
s
Stable distribution:
0  = 1 
1  = 2 2
…
s-1  = s s
Zvi Wiener
n
n

n!  n
n
!
 n  s i  s i , where  





i
i
!

i 0
i  0 i!
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Probabilities
n
 n
e
 (1  s )
n
!
n  s i 
(1  n)(1  s,  )


i  0 i!
• Probability of losing a new project due to
capital requirements is equal to the probability
of being in state s, i. e. s.
• Termination of projects does not have to be
Poissonian, only mean and variance matter.
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Expected Profit
s( s, )
E( profit )   1    s( p) p  p  
p
(1  s, )
An optimal p (risk measurement system) can
be found by maximizing the expected profit
stream.
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slide 51
Example
k ( p)  0.015 (0.08  0.015)e

p
q
• Capital requirement as a function of p
(price) and q (scaling factor), varies between
1.5% and 8%.
Zvi Wiener
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slide 52
Example
k ( p)  0.015 (0.08  0.015)e
Amax14.25

p
q
q=0.5
14
13.75
q=1
13.5
q=3
13.25
1
2
3
4
p
12.75
12.5
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slide 53
Example of a bank
• Capital $200M
• Average project is $20K
• On average 200 new projects arrive each day
• Average life of a project is 2 years
• 15% of assets are traded and q=1
• spread =1.25%
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CF4
slide 54
Expected profit
33
32.5
32
31.5
31
30.5
1
2
3
4
5
6
rent p
29.5
Bank’s profit as a function of cost p.
C=$200M, arrival rate 200/d,
size $20K, average life 2 yr.,
spread 1.25%, q=1, 15% of assets are traded.
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CF4
slide 55
Expected profit
25
24
23
22
21
1
2
3
4
5
6
rent p
Bank’s profit as a function of cost p.
C=$200M, arrival rate 200/d,
size $20K, average life 2 yr.,
spread 1%, q=1, 5% of assets are traded.
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slide 56
Conclusion
• Expensive systems are appropriate for banks
with
• low capitalization
• operating in an unstable environment
• Cheaper methods (like the standard approach)
should be appropriate for banks with
• high capitalization
• small trading book
• operating in a stable environment
• many small uncorrelated, long living projects
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CF4
slide 57
A simple intuitive and flexible model of
optimal choice of risk measuring method.
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CF4
slide 58